Simplify powers using prime-factorization rules Compute the exact value of (9^2 * 18^4) / 3^16.

Introduction / Context:Exponent problems with composite bases are best handled by prime factorization. Converting all terms to powers of the same primes allows straightforward cancellation.

Given Data / Assumptions:

• Expression: (9^2 * 18^4) / 3^16.
• 3 is the key prime in all bases.

Concept / Approach:Rewrite 9 and 18 in terms of 3 (and 2 for 18). Use a^m * a^n = a^(m+n) and a^m / a^n = a^(m−n). Keep factors of 2 separate.

Step-by-Step Solution:9 = 3^2 ⇒ 9^2 = (3^2)^2 = 3^4.18 = 2 * 3^2 ⇒ 18^4 = 2^4 * (3^2)^4 = 2^4 * 3^8.Numerator: 3^4 * (2^4 * 3^8) = 2^4 * 3^(4+8) = 2^4 * 3^12.Divide by 3^16 ⇒ 2^4 * 3^(12−16) = 2^4 * 3^(−4) = 16 / 81.

Verification / Alternative check:Compute via exponent subtraction directly or evaluate numerically to confirm ≈ 0.1975… equals 16/81.

Why Other Options Are Wrong:2/3, 4/9, 32/243, 8/27 are different reductions that do not match the exact exponent arithmetic here.

Common Pitfalls:Forgetting 18 includes 2; mis-adding exponents; dividing powers incorrectly.

Final Answer:16/81